The simple question of who invented the digital computer has many possible answers. Several strands of development — both theoretical and practical — converged around 1950, and no single inventor can be identified. Before that time, numerous mechanical calculating machines were devised and constructed. These greatly facilitated the preparation of mathematical tables, crucial for engineering … Continue reading Who Invented the Digital Computer? There’s No Simple Answer.
Author: thatsmaths
The Hindu-Arabic Numerals: a Blessing for Modern Civilisation
Directly opposite the RDS in Dublin, on the site of the former AIB headquarters, stand two impressive new office blocks separated by a plaza, Fibonacci Place. Leonardo Bonacci — known as Fibonacci, the son of Bonaccio — lived from 1170 to about 1245 AD. It is he who we must thank for bringing us the … Continue reading The Hindu-Arabic Numerals: a Blessing for Modern Civilisation
Music and Maths are Inextricably Intertwined.
A crucial link between mathematics and music is nicely illustrated by the Tonnetz, a geometric diagram representing the harmonic relationships between the notes of the musical scale. An early version of the Tonnetz appeared in Leonhard Euler’s book Tentamen novae theoriae musicae (A new theory of music), published in 1739 [TM274 or search for “thatsmaths” at irishtimes.com]. A modern … Continue reading Music and Maths are Inextricably Intertwined.
John von Neumann: Computer Wizard and AI Visionary
With unique and outstanding achievements in mathematics and science, John von Neumann was on an intellectual level far above us; the oft-misused term genius truly applies to him. Von Neumann, brilliant in both pure and applied mathematics, was a towering figure of 20th century science. As a child, he demonstrated a prodigious memory and computational … Continue reading John von Neumann: Computer Wizard and AI Visionary
Finding a Horseshoe on the Beaches of Rio
In 1966, American mathematician Steve Smale was awarded a Fields Medal, a kind of Nobel Prize for mathematics. At a press conference at the International Congress of Mathematicians in Moscow, Smale attacked both US and Russian foreign policy. He was vehemently opposed to military aggression by the two super-powers. His pacifist position resulted in serious … Continue reading Finding a Horseshoe on the Beaches of Rio
Going for Gold: Henri Poincaré and Solar System Stability
In his classical textbook on mechanics, Edmund Whittaker described the three-body problem as “the most celebrated of all dynamical problems”. From 1906 to 1912, Whittaker was Andrews Professor at Trinity College Dublin and Royal Astronomer of Ireland. The three-body problem is to determine the motion of three massive bodies moving in space under mutual gravitational … Continue reading Going for Gold: Henri Poincaré and Solar System Stability
The Future of Physics may be Surreal
Imagine the Earth were to shrink to the size of a marble. We might be in trouble, but the planet would continue its smooth course around the Sun while the Moon would maintain its orbit, circling Earth once a month. Isaac Newton proved that the Earth’s gravitational pull would be the same even if all … Continue reading The Future of Physics may be Surreal
The James Webb Telescope: Viewing the Universe from Lagrange Point L2
There are five sweet spots where a spacecraft can keep pace with Earth as both orbit the Sun. They are called the Lagrange points, after the brilliant French mathematician Joseph-Louis Lagrange who found special solutions to what is called the “three-body problem”. To locate the second Lagrange point, L2, draw a line 150 million km … Continue reading The James Webb Telescope: Viewing the Universe from Lagrange Point L2
Euler’s Identity: the Most Beautiful Equation in Mathematics
A recent entry in the visitor’s book of the James Joyce Tower & Museum in Sandycove, Dublin perplexed the Friends of Joyce’s Tower, the volunteers who run the museum. The entry, reproduced as the boxed equation in the photograph, seemed as impenetrable as a passage of Finnegans Wake and quite beyond decryption. When invited to … Continue reading Euler’s Identity: the Most Beautiful Equation in Mathematics
The Emergence of Spring
At 9.01AM today, the Sun crosses the Earth’s equator, marking the vernal equinox, the date when the direction of sunrise is due east and sunset due west. The Earth’s axis is at right angles to the line from Sun to Earth, the Sun is directly over the equator and the lengths of day and night … Continue reading The Emergence of Spring
Quantum Physics, a Century Old, still Passes Understanding
This is the International Year of the Quantum, celebrating quantum science and its many applications. This year marks the centenary of Werner Heisenberg’s seminal work in modern physics. In recent years, there have been dramatic developments in quantum communications, cryptography and computing, and quantum science is key to solving pressing problems in clean energy, climate … Continue reading Quantum Physics, a Century Old, still Passes Understanding
Surprising Discoveries in the Family Tree of Mathematics
The German word Doktorvater for a doctoral advisor indicates the close relationship between a PhD student and his or her supervisor. The relationship is often pivotal in determining the future career of the student, and the advisor also gains much from the interaction. Just as a genealogical tree can reveal fascinating information, a mathematical family … Continue reading Surprising Discoveries in the Family Tree of Mathematics
The Surprising Utility of Square Wheels
The London Eye, a vast observation wheel with a diameter of 120 metres, on the south bank of the River Thames, opened in 2000. Resembling an enormous bicycle wheel, it has 32 passenger capsules, each weighing ten tons, attached to its circumference. Each capsule holds 25 people, so up to 800 may be on the … Continue reading The Surprising Utility of Square Wheels
Fermi Resonance and Climate Change
Carbon dioxide (CO2) is the main culprit in causing our climate to change, but it is only recently that a key property of CO2 has been elucidated. This is linked to a surprising interaction between the molecular vibrations of carbon dioxide: the various modes of vibration of the molecules resonate with each other in a … Continue reading Fermi Resonance and Climate Change
The Never-ending Quest for Enormous Prime Numbers
The ongoing search for ever-larger prime numbers continues apace. Primes are the atoms of arithmetic: every whole number is a unique product of primes. For example, 21 is the product of primes 3 and 7, while 23 is itself prime. The primes play a central role in pure mathematics, in the field of number theory, … Continue reading The Never-ending Quest for Enormous Prime Numbers
Joining Forces to Improve Weather Forecasts
Since 1950, there has been a quiet but steady revolution in meteorology, and especially in numerical weather prediction (NWP). The growth in accuracy, range and scope of weather forecasts over the past half-century has been spectacular. As late as the mid-1970s, forecasts were seriously unreliable. The diagram illustrates the inexorable increase in skill of the … Continue reading Joining Forces to Improve Weather Forecasts
Hamilton’s Dynamics: a Prescient Framework for Quantum Mechanics
While mathematics may be viewed as an abstract creation, its origins lie in the physical world. The need to count animals and share food supplies led to the development of the concept of numbers. With five-fingered hands, we naturally tended to count in tens. Arithmetic methods were needed to allocate land, organize armies and calculate … Continue reading Hamilton’s Dynamics: a Prescient Framework for Quantum Mechanics
Progress towards a Grand Unified Theory of Mathematics
Science advances by overturning theories, replacing them by better ones. Sometimes, the old theories continue to serve as valuable approximations, as with Newton’s laws of motion [TM260 or search for “thatsmaths” at irishtimes.com]. Sometimes, the older theories become redundant and are forgotten. The theory of phlogiston, a fire-like element released during combustion, and the luminiferous … Continue reading Progress towards a Grand Unified Theory of Mathematics
The Doppler Effect: Simple but Remarkably Useful
We have all noticed how the horn of a speeding car changes as it approaches: each wave-peak is emitted from a closer point, so the wave is “squeezed” and the pitch increases. As the car recedes, the reverse effect stretches the wave, making it sound lower. The changing pitch of the note is called the … Continue reading The Doppler Effect: Simple but Remarkably Useful
To See a World in a Grain of Sand
Gaze up at the night sky and you catch a glimpse of infinity. The stars seem numberless; the immensity of the universe is profoundly impressive, leading us to wonder about the nature of our cosmic home. Is space finite or unlimited in extent? If it is bounded, what lies outside? The notion of infinity is … Continue reading To See a World in a Grain of Sand
Mathematical Fun and Games at Maison Poincaré
This article comes from the Institut Henri-Poincaré (IHP), part of Sorbonne University, in the Latin Quarter of Paris. IHP has no permanent researchers but serves as a venue for mathematical collaborations and organises a series of programmes, seminars, lectures and training courses, welcoming more than 10,000 mathematicians each year. The institute also publishes four international … Continue reading Mathematical Fun and Games at Maison Poincaré
Resonant Vibrations from Atoms to the Far Horizons of the Cosmos
Panta Rhei — everything flows — said Heraclites, describing the impermanence of the world. He might well have said “everything vibrates”. From sub-atomic particles to the farthest reaches of the cosmos we find oscillations. Vibration is key for aircraft wing, motor engine and optical system design. Ocean tides forced by the Moon and seasonal variations … Continue reading Resonant Vibrations from Atoms to the Far Horizons of the Cosmos
Pythagorean Tuning and the Spiral of Theodorus
Most of us are familiar with the piano keyboard. There are twelve distinct notes in each octave, or thirteen if we include the note completing the chromatic scale. The illustration above shows a complete scale from middle C (or C$latex {_4}&fg=000000$) to the C above (or C$latex {_5}&fg=000000$). Eight of the notes --- C, D, … Continue reading Pythagorean Tuning and the Spiral of Theodorus
The Many Schools of Mathematical Thought
Mathematics is used widely, playing a central role in science and engineering and, increasingly, in the social and biological sciences. But users seldom consider the fundamental nature of mathematics. Many cannot improve on the vacuous definition: mathematics is what is done by mathematicians. We could try harder, with something like “mathematics is the language of … Continue reading The Many Schools of Mathematical Thought
Breaking a Stick to Form a Triangle
We consider a simple problem in probability: A thin rod is broken at random into three pieces. What is the probability that these three pieces can be used to form a triangle? This problem is solved without difficulty. If the rod is of unit length, aligned along the interval $latex {0\le x \le 1}&fg=000000$, we … Continue reading Breaking a Stick to Form a Triangle
Mileva Marić and the Special Theory of Relativity
The year 1905 was Albert Einstein’s “miracle year”. In that year, he published four papers in the renowned scientific journal Annalen der Physik. The first, on the photoelectric effect, established the quantum nature of light, and led to the award of a Nobel Prize some 17 years later. The second, on Brownian motion, confirmed the … Continue reading Mileva Marić and the Special Theory of Relativity
ENSO: The Oscillating Atmosphere and Ocean
The weather of 2023 was certainly interesting, with broken records in Ireland and around the world. Newspaper articles attributed the cause of the heat waves, droughts, floods and fires to the climate pattern known as El Niño. Less restrained reports claimed that this year’s weather would be even more anomalous [TM253 or search for “thatsmaths” … Continue reading ENSO: The Oscillating Atmosphere and Ocean
Post600: New Year’s Greetings, 2024
The That’s Maths blog has been active since July 2012. This is post number 600. For the past eleven and a half years, there has been a post every Thursday. For a complete list in chronological order, just press the “Contents” button, or click here. The posts have covered a wide range of topics in … Continue reading Post600: New Year’s Greetings, 2024
Rubik’s Cube Solvable in 20 Moves
In 1974, the brilliant Hungarian professor of architecture, Ernö Rubik, invented the puzzle that has made his name familiar all over the world. When it was mass-produced, from 1980 onwards, the cube became an international craze; vast numbers were made and they brought both great fun and great frustration to millions of children of all … Continue reading Rubik’s Cube Solvable in 20 Moves
The Decline of the Mayans: a Warning Signal for Us
The rising temperatures of today’s climate are being linked to extreme weather, droughts, floods and intense storms, and global food and water supplies are coming under severe stress. While the current changes are unprecedented in their rapidity, climate variations in the past have had devastating consequences. What can we learn from them? [TM252 or search for … Continue reading The Decline of the Mayans: a Warning Signal for Us
The Sieve of Eratosthenes and a Partition of the Natural Numbers
The sieve of Eratosthenes is a method for finding all the prime numbers less than some maximum value $latex {M}&fg=000000$ by repeatedly removing multiples of the smallest remaining prime until no composite numbers less than or equal to $latex {M}&fg=000000$ remain. The sieve provides a means of partitioning the natural numbers. We examine this partition … Continue reading The Sieve of Eratosthenes and a Partition of the Natural Numbers
The Logistic Map is hiding in the Mandelbrot Set
The logistic map is a simple second-order function on the unit interval: $latex \displaystyle x_{n+1} = r x_n (1-x_n) \,, &fg=000000$ where $latex {x_n}&fg=000000$ is the variable value at stage $latex {n}&fg=000000$ and $latex {r}&fg=000000$ is the ``growth rate''. For $latex {1 \le r \le 4}&fg=000000$, the map sends the unit interval [0,1] into itself. … Continue reading The Logistic Map is hiding in the Mandelbrot Set
The Golden Key to Riemann’s Hypothesis
The Riemann Hypothesis Perhaps the greatest unsolved problem in mathematics is to explain the distribution of the prime numbers. The overall ``thinning out'' of the primes less than some number $latex {N}&fg=000000$, as $latex {N}&fg=000000$ increases, is well understood, and is demonstrated by the Prime Number Theorem (PNT). In its simplest form, PNT states that … Continue reading The Golden Key to Riemann’s Hypothesis
Sharkovsky’s Theorem
This post is an extension and elaboration of two recent posts, with more technical details ``The reasonable man adapts himself to the world: the unreasonable one persists in trying to adapt the world to himself. Therefore all progress depends on the unreasonable man.'' … Continue reading Sharkovsky’s Theorem
Oleksandr Sharkovsky and Chaos Theory
We all know how a simple action at a critical moment can change our lives. Over the past half-century, with the growing evidence of how small changes can lead to dramatic developments, there has been a paradigm shift in science. Earlier attempts to predict the future as if it were determined with certainty have given … Continue reading Oleksandr Sharkovsky and Chaos Theory
The Logistic Map: a Simple Model with Rich Dynamics
Suppose the population of the world $latex {P(t)}&fg=000000$ is described by the equation $latex \displaystyle \frac{\mathrm{d}P} {\mathrm{d}t} = a P \,. &fg=000000$ Then $latex {P(t)}&fg=000000$ grows exponentially: $latex {P(t) = P_0 \exp(at)}&fg=000000$. This was the nightmare prediction of Thomas Robert Malthus. Taking a value $latex {a=0.02\ \mathrm{yr}^{-1}}&fg=000000$ for the growth rate, we get a doubling … Continue reading The Logistic Map: a Simple Model with Rich Dynamics
Yin and Yang — and East and West
The duality encapsulated in the concept of yin-yang is at the origin of many aspects of classical Chinese science and philosophy. Many dualities in the natural world --- light and dark, fire and water, order and chaos --- are regarded as physical manifestations of this duality. Yin is the receptive and yang the active principle. … Continue reading Yin and Yang — and East and West
The Axiom of Choice: Shoes & Socks and Non-constructive Proofs
Recall Euclid's proof that there is no limit to the list of prime numbers. One way to show this is that, by assuming that some number $latex {p}&fg=000000$ is the largest prime, we arrive at a contradiction. The idea is simple yet powerful. A Non-constructive Proof Suppose $latex {p}&fg=000000$ is prime and there are no … Continue reading The Axiom of Choice: Shoes & Socks and Non-constructive Proofs
Earth’s Digital Twins can help us to avert Disaster
Imagine another Earth, just like ours, but running a year ahead. Observing it, we could foretell events over the coming weeks or months, and take action to avoid catastrophes. There is no such planet! Even if there were, conditions there would diverge rapidly from ours, so it would provide no guidance on our future. But … Continue reading Earth’s Digital Twins can help us to avert Disaster
Elusive Transcendentals
The numbers are usually studied in layers of increasing subtlety and intricacy. We start with the natural, or counting, numbers $latex {\mathbb{N} = \{ 1, 2, 3, \dots \}}&fg=000000$. Then come the whole numbers or integers, $latex {\mathbb{Z} = \{ \dots, -2, -1, 0, 1, 2, \dots \}}&fg=000000$. All the ratios of these (avoiding division … Continue reading Elusive Transcendentals
Digital Signatures using Edwards Curves
A digital signature is a mathematical means of verifying that an e-document is authentic, that it has come from the claimed sender and that it has not been tampered with or corrupted during transit. Digital signatures are a standard component of cryptographic systems. They use asymetric cryptography that is based on key pairs, consisting of … Continue reading Digital Signatures using Edwards Curves
A Remarkable Sequence in OEIS
The On-Line Encyclopedia of Integer Sequences (OEIS), launched in 1996, now contains 360,000 entries. It attracts a million visits a day, and has been cited about 10,000 times. It is now possible for anyone in the world to propose a new sequence for inclusion in OEIS. The goal of the database is to include all … Continue reading A Remarkable Sequence in OEIS
Maths in Action at the Rugby World Cup
On September 8, I opened The Irish Times to find an A2 Poster with the programme for the Rugby World Cup. The plan showed the twenty teams who must do battle in which ultimate triumph requires survival through the preliminary rounds and victory in quarter-finals, semi-finals and the final climax. We have reached the quarter-finals … Continue reading Maths in Action at the Rugby World Cup
A Memorable Memo: Responding to Over-assiduous Administrators
Anyone who has worked in a large organization, with an over-loaded Administration Division, will sympathise with the actions of two scientists at the Los Alamos National Laboratory (LANL) in issuing a spoof Memorandum. They had become frustrated with the large number of mimeographed notes circulated by Administration and Services, or A&S, ``to keep laboratory members … Continue reading A Memorable Memo: Responding to Over-assiduous Administrators
Hamilton’s Semaphore Code and Signalling System
Sir William Rowan Hamilton (1805-1865) was Ireland's most ingenious mathematician. When he was just fifteen years old, Hamilton and a schoolfriend invented a semaphore-like signalling system. On 21 July 1820, Hamilton wrote in his journal how he and Tommy Fitzpatrick set up a mark on a tower in Trim and were able to view it … Continue reading Hamilton’s Semaphore Code and Signalling System
Sixth Irish History of Mathematics (IHoM) Conference
I attended the sixth conference of the Irish History of Mathematics (IHoM) group at Maynooth University yesterday (Wednesday 30th August 2023). What follows is a personal summary of the presentations. This summary has no official status. If speakers or attendees spot any errors, please let me know and I will correct them. [1] After a … Continue reading Sixth Irish History of Mathematics (IHoM) Conference
Retroreflectors: Right Angles Save Lives
As everyone knows, left and right are swapped in a mirror image. Or are they? It is really front and back that are reversed, but that's a story for another day. During a visit to the Science and Industry Museum in Paris some years ago, I stood facing a spinning mirror. Lifting one arm, I … Continue reading Retroreflectors: Right Angles Save Lives
Music and Maths from Bach to Bacherach
At a fundamental level, music may be described as a train of vibrations in the air. It can be further reduced to numbers — a string of binary digits (bits) that can be stored on a CD or sent around the world in a split second. But music carries enormous emotional content and can stir … Continue reading Music and Maths from Bach to Bacherach
Proofs without Words
The sum of the first $latex {n}&fg=000000$ odd numbers is equal to the square of $latex {n}&fg=000000$: $latex \displaystyle 1 + 3 + 5 + \cdots + (2n-1) = n^2 \,. &fg=000000$ We can check this for the first few: $latex {1 = 1^2,\ \ 1+3=2^2,\ \ 1+3+5 = 3^2}&fg=000000$. But how do we prove … Continue reading Proofs without Words
Maths in the Time of the Pharaohs
Why would the Ancient Egyptians have any interest in or need for mathematics? There are many reasons. They had a well-organised and developed civilisation extending over millennia. Science and maths must have played important or even essential roles in this culture. They needed measurement for land surveying and for designing irrigation canals, arithmetic for accounting … Continue reading Maths in the Time of the Pharaohs
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